Chocolate covered candy (noch nicht übersetzt)

Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation: $b^2 x^2 + b^2 y^2 + a^2 z^2 = a^2 b^2$.

Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one millimeter thick.

If $a = 1$ mm and $b = 1$ mm, the amount of chocolate required is $\dfrac{28}{3} \pi$ mm³

If $a = 2$ mm and $b = 1$ mm, the amount of chocolate required is approximately 60.35475635 mm³.

Find the amount of chocolate in mm³ required if $a = 3$ mm and $b =1$ mm. Give your answer as the number rounded to 8 decimal places behind the decimal point.